Olympiad problems

2010 AMC 10, 9

Tags: AMC , AMC 12 , MATHCOUNTS , AIME , algebra , polynomial , Vieta

A [i]palindrome[/i], such as $ 83438$, is a number that remains the same when its digits are reversed. The numbers $ x$ and $ x \plus{} 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x? $ \textbf{(A)}\ 20\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 23\qquad \textbf{(E)}\ 24$

1981 IMO Shortlist, 12

Tags: algebra , polynomial , Vieta , number theory , IMO , IMO 1981 , Diophantine equation

Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

2003 Romania Team Selection Test, 7

Tags: algebra , polynomial , Vieta , trigonometry , algebra proposed

Find all integers $a,b,m,n$, with $m>n>1$, for which the polynomial $f(X)=X^n+aX+b$ divides the polynomial $g(X)=X^m+aX+b$. [i]Laurentiu Panaitopol[/i]

2007 ITest, 18

Tags: Vieta , quadratics

Suppose that $x^3+px^2+qx+r$ is a cubic with a double root at $a$ and another root at $b$, where $a$ and $b$ are real numbers. If $p=-6$ and $q=9$, what is $r$? $\textbf{(A) }0\hspace{20.2em}\textbf{(B) }4$ $\textbf{(C) }108\hspace{19.3em}\textbf{(D) }\text{It could be 0 or 4.}$ $\textbf{(E) }\text{It could be 0 or 108.}\hspace{12em}\textbf{(F) }18$ $\textbf{(G) }-4\hspace{19em}\textbf{(H) } -108$ $\textbf{(I) }\text{It could be 0 or }-4.\hspace{12em}\textbf{(J) }\text{It could be 0 or }-108.$ $\textbf{(K) }\text{It could be 4 or }-4.\hspace{11.5em}\textbf{(L) }\text{There is no such value of }r.$ $\textbf{(M) }1\hspace{20em}\textbf{(N) }-2$ $\textbf{(O) }\text{It could be }-2\text{ or }-4.\hspace{10.3em}\textbf{(P) }\text{It could be 0 or }-2.$ $\textbf{(Q) }\text{It could be 2007 or a yippy dog.}\hspace{6.6em}\textbf{(R) }2007$

2012 Online Math Open Problems, 25

Tags: Online Math Open , algebra , polynomial , function , Vieta

Let $a,b,c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7$. Given that $P$ is a cubic polynomial such that $P(a)=b+c$, $P(b) = c+a$, $P(c) = a+b$, and $P(a+b+c) = -16$, find $P(0)$. [i]Author: Alex Zhu[/i]

2000 AMC 10, 24

Tags: function , Vieta , algebra , polynomial , AMC

Let $f$ be a function for which $f\left(\frac x3\right)=x^2+x+1$. Find the sum of all values of $z$ for which $f(3z)=7$. $\text{(A)}\ -\frac13\qquad\text{(B)}\ -\frac19 \qquad\text{(C)}\ 0 \qquad\text{(D)}\ \frac59 \qquad\text{(E)}\ \frac53$

2024 Harvard-MIT Mathematics Tournament, 1

Tags: algebra , Vieta

Suppose $r$, $s$, and $t$ are nonzero reals such that the polynomial $x^2 + rx + s$ has $s$ and $t$ as roots, and the polynomial $x^2 + tx + r$ has $5$ as a root. Compute $s$.

1968 AMC 12/AHSME, 13

Tags: quadratics , Vieta , AMC

If $m$ and $n$ are the roots of $x^2+mx+n=0$, $m\ne0$, $n\ne0$, then the sum of the roots is: $\textbf{(A)}\ -\dfrac{1}{2} \qquad \textbf{(B)}\ -1 \qquad \textbf{(C)}\ \dfrac{1}{2} \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \text{Undetermined} $

2014 Cezar Ivănescu, 3

Tags: Vieta , linear algebra

Let $ A,B,C,D $ be four $ 2\times 2 $ complex matrices such that $ A-D $ is invertible and such that $$ A^2+BA+C=0=D^2+BD+C. $$ Prove that $ \text{tr} (A+D) =-\text{tr} B $ and $ \det (AD) =\det C. $

1988 IMO, 1

Tags: algebra , polynomial , Vieta , Inequality , roots , IMO , IMO 1988

Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

1985 IberoAmerican, 3

Tags: Columbia , Vieta , algebra unsolved , algebra

Find all the roots $ r_{1}$, $ r_{2}$, $ r_{3}$ y $ r_{4}$ of the equation $ 4x^{4}\minus{}ax^{3}\plus{}bx^{2}\minus{}cx\plus{}5 \equal{} 0$, knowing that they are real, positive and that: \[ \frac{r_{1}}{2}\plus{}\frac{r_{2}}{4}\plus{}\frac{r_{3}}{5}\plus{}\frac{r_{4}}{8}\equal{} 1.\]

2002 AMC 10, 10

Tags: quadratics , Vieta

Suppose that $ a$ and $ b$ are are nonzero real numbers, and that the equation $ x^2\plus{}ax\plus{}b\equal{}0$ has solutions $ a$ and $ b$. Then the pair $ (a,b)$ is $ \textbf{(A)}\ (\minus{}2,1) \qquad \textbf{(B)}\ (\minus{}1,2) \qquad \textbf{(C)}\ (1,\minus{}2) \qquad \textbf{(D)}\ (2,\minus{}1) \qquad \textbf{(E)}\ (4,4)$

2008 ITest, 53

Tags: algebra , polynomial , Vieta

Find the sum of the $2007$ roots of \[(x-1)^{2007}+2(x-2)^{2006}+3(x-3)^{2005}+\cdots+2006(x-2006)^2+2007(x-2007).\]

2017 NIMO Summer Contest, 9

Tags: algebra , polynomial , Vieta

Let $P$ be a cubic monic polynomial with roots $a$, $b$, and $c$. If $P(1)=91$ and $P(-1)=-121$, compute the maximum possible value of \[\dfrac{ab+bc+ca}{abc+a+b+c}.\] [i]Proposed by David Altizio[/i]

1971 AMC 12/AHSME, 20

Tags: algebra , polynomial , Vieta , absolute value , AMC

The sum of the squares of the roots of the equation $x^2+2hx=3$ is $10$. The absolute value of $h$ is equal to $\textbf{(A) }-1\qquad\textbf{(B) }\textstyle\frac{1}{2}\qquad\textbf{(C) }\textstyle\frac{3}{2}\qquad\textbf{(D) }2\qquad \textbf{(E) }\text{None of these}$

2009 CHKMO, 2

Tags: function , Euler , algebra , polynomial , Vieta , quadratics , number theory

Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.

2011 NIMO Summer Contest, 9

Tags: algebra , polynomial , Vieta

The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$. [i]Proposed by Eugene Chen [/i]

1960 AMC 12/AHSME, 11

Tags: calculus , integration , Vieta , AMC

For a given value of $k$ the product of the roots of \[ x^2-3kx+2k^2-1=0 \] is $7$. The roots may be characterized as: $ \textbf{(A) }\text{integral and positive} \qquad\textbf{(B) }\text{integral and negative} \qquad$ $\textbf{(C) }\text{rational, but not integral} \qquad\textbf{(D) }\text{irrational} \qquad\textbf{(E) } \text{imaginary} $

1995 South africa National Olympiad, 2

Tags: algebra , polynomial , Vieta , function , number theory unsolved , number theory

Find all pairs $(m,n)$ of natural numbers with $m<n$ such that $m^2+1$ is a multiple of $n$ and $n^2+1$ is a multiple of $m$.

2009 Harvard-MIT Mathematics Tournament, 5

Tags: algebra , polynomial , Vieta , sum of roots

Let $a$, $b$, and $c$ be the $3$ roots of $x^3-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}.$

1978 AMC 12/AHSME, 13

Tags: algebra , polynomial , Vieta , AMC

If $a,b,c,$ and $d$ are non-zero numbers such that $c$ and $d$ are the solutions of $x^2+ax+b=0$ and $a$ and $b$ are the solutions of $x^2+cx+d=0$, then $a+b+c+d$ equals $\textbf{(A) }0\qquad\textbf{(B) }-2\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad \textbf{(E) }(-1+\sqrt{5})/2$

1961 AMC 12/AHSME, 29

Tags: quadratics , algebra , polynomial , Vieta , AMC

Let the roots of $ax^2+bx+c=0$ be $r$ and $s$. The equation with roots $ar+b$ and $as+b$ is: $ \textbf{(A)}\ x^2-bx-ac=0$ $\qquad\textbf{(B)}\ x^2-bx+ac=0$ $\qquad\textbf{(C)}\ x^2+3bx+ca+2b^2=0$ ${\qquad\textbf{(D)}\ x^2+3bx-ca+2b^2=0 }$ ${\qquad\textbf{(E)}\ x^2+bx(2-a)+a^2c+b^2(a+1)=0} $

2010 Contests, 2

Tags: inequalities , analytic geometry , conics , parabola , algebra , polynomial , Vieta

Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2\minus{}ax\plus{}1\equal{}0$ we build the sequence with $ S_{n}\equal{}x_{1}^n \plus{} x_{2}^n$. [b]a)[/b]Prove that the sequence $ \frac{S_{n}}{S_{n\plus{}1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing. [b]b)[/b]Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}\plus{}\cdots \plus{}\frac{S_{n}}{S_{n\plus{}1}}>n\minus{}1$

1989 India National Olympiad, 2

Tags: algebra , polynomial , Vieta , algebra unsolved

Let $ a,b,c$ and $ d$ be any four real numbers, not all equal to zero. Prove that the roots of the polynomial $ f(x) \equal{} x^{6} \plus{} ax^{3} \plus{} bx^{2} \plus{} cx \plus{} d$ can't all be real.

PEN B Problems, 5

Tags: modular arithmetic , algebra , polynomial , Vieta , symmetry , search , induction

Let $p$ be an odd prime. If $g_{1}, \cdots, g_{\phi(p-1)}$ are the primitive roots $\pmod{p}$ in the range $1<g \le p-1$, prove that \[\sum_{i=1}^{\phi(p-1)}g_{i}\equiv \mu(p-1) \pmod{p}.\]